Abstract
The robust fusion filtering problem is considered for linear time-varying uncertain systems observed by multiple sensors. A performance index function for this problem is defined as an indefinite quadratic inequality which is solved by the projection method in Krein space. On this basis, a robust centralized finite horizon fusion filtering algorithm is proposed. However, this centralized fusion method is with poor real time property, as the number of sensors increases. To resolve this difficulty, within the sequential fusion framework, the performance index function is described as a set of quadratic inequalities including an indefinite quadratic inequality. And a sequential robust finite horizon fusion filtering algorithm is given by solving this quadratic inequality group. Finally, two simulation examples for time-varying/time-invariant multisensor systems are exploited to demonstrate the effectiveness of the proposed methods in the respect of the real time property and filtering accuracy.
1. Introduction
In many advanced systems, multiple homogeneous or heterogeneous sensors are spatially distributed to provide large coverage, diverse viewing angles of the things of interest [1, 2]. How to deal with large amounts of overlapping and complementary data sampled by these sensors is a crucial issue. The fusion filtering technology could effectively integrate these data to estimate the signal of interest.
In the existing literature, the research of fusion filter has already become a focus in recent years. Most of the existing results are usually developed from Kalman filter [3–7], on the basis of two necessary assumptions: the system parameters are given, and the system noises satisfy the Gaussian distributions with given statistic characteristics. These assumptions, however, are usually too idealistic to obtain in practice. Recently, several fusion filtering methods are also developed with different assumptions. While the system parameters include some uncertainties and the system noises are Gaussian, by describing the system parameter uncertainties as multiplicative noises or the norm-bounded uncertainties, several robust Kalman fusion filters are deduced in [8–10]. While the system parameters are given and the statistic characteristic of system noises is unknown, some centralized fusion filters are presented based on the linear matrix inequality (LMI) technology or the Riccati equation technology [11–13]. For the linear time-invariant multisensor system with energy-bounded noises and norm-bounded uncertain parameters, the centralized robust fusion filters are proposed in [14, 15]. In [16], a centralized distributed -consensus filtering method is proposed for discrete time-varying systems by solving a set of different linear matrix inequalities in each filtering period and further extended for two kinds of uncertain systems.
However, there are still some performance and application deficiencies in the (robust) fusion filters mentioned above. The deficiencies on performance are mainly embodied in the real time property. This is due to the centralized fusion structure of these filters [11–15], in which the measurement functions of different sensors are augmented into a high-dimensional measurement function, whose dimension increases with the increase of sensors. Therefore, the running time of these fusion filters can be affected by the implicit high-dimensional operation. What is more, these fusion filters are designed on the basis of the measurements sampled by all sensors. It is implied that the estimate of the signal to be estimated cannot be obtained until all measurements sampled by different sensors in a fusion period arrive at the information processer. Particularly when measurements are transmitted with random delayed phenomenon, the real time performance of the centralized fusion methods is usually very poor. The deficiency on application refers to the fact that most available literature concerning the fusion filtering problems has been limited to time-invariant systems, and the state estimation problem for the corresponding time-varying systems has not been paid adequate research attention to despite its clear engineering significance.
In this paper, we aim to investigate the robust fusion filtering method for time-varying multisensor uncertain systems. The research work in this paper mainly includes the following parts:(i)In this paper, the impacts of the parameter uncertainty and the system noises on the fusion estimate errors are expressed by an indefinite quadratic inequality, whose stationary can be given by a projection method in Krein space. On this basis, a robust centralized finite horizon fusion filtering algorithm is designed.(ii)In order to improve the real time property of the above robust fusion filtering algorithm, the performance index function is reformulated into a set of quadratic inequalities. By sequentially solving these quadratic inequalities, a real time robust finite horizon fusion filtering algorithm is developed.
The remainder of this paper is organized as follows. In Section 2, the time-varying multisensor system is formulated. Two robust finite horizon fusion filtering algorithms are proposed in Section 3, respectively, according to the centralized fusion strategy and the sequential one. Simulation results and comparisons are presented in Section 4, and some conclusions are given in Section 5.
Notation. The notation used here is fairly standard except where otherwise stated. The superscript “” stands for matrix transposition, denotes the -dimensional Euclidean space, and denotes the identity matrix with appropriate dimension. The notation , where is positive definite. The vectors in Hilbert space are denoted by bold face letters, such as “,” and the ones in Krein space are denoted by the bold face letters with bar, such as “.” stands for the inner product in Krein space.
2. Problem Formulation
Consider the following time-varying -sensor system with uncertain parameters:where is the state vector. is the measurement output of sensor . is the process noise and is the corresponding measurement noise of sensor . is the signal to be estimated. , , are the given matrices with compatible dimensions. is a real-valued uncertain matrix satisfying , in which , are known time-varying matrices and is time-varying uncertainty satisfying .
3. Robust Finite Horizon Fusion Filtering Algorithms
Define the following auxiliary variables [17]:Then the system shown in (1) can be rewritten as
Due to , , we can also get
Denote as the fusion filtering result of , based on the measurements , and . The transfer function of the system noises and can be expressed aswhere and is an initial estimate of . is a given positive definite matrix.
For a given scalar , the following constraint is given to map the system noises to the filtering error:It is obvious from (6) thatCombining the above constraints (4) and (7), we can obtain the following performance index function for the robust fusion filtering process: Rewrite (8) as
The above performance index means(1)there is a minimum of at a stationary point ;(2)the minimum .
The stationary point of indefinite quadratic forms in Hilbert space corresponds to a projection in Krein space [17–20]. In the remainder of this section, the projections in Krein space will be solved to obtain the stationary point of and further to yield the estimates of the signal to be estimated.
3.1. Centralized Robust Finite Horizon Fusion Filtering Algorithm
Define the following augmented matrices: Then we can rewrite (1) in the following augmented matrix form:here . And the performance index function shown in (9) can be further expressed as in which .
Theorem 1. Given a positive scalar , for the augmented matrix system shown in (11), the following robust finite horizon fusion filter can be given to satisfy the performance index function (9), based on the centralized fusion strategy:whereThe existing condition of this robust fusion filter is that and have the same inertia index.
Proof. The performance index function (9) can be expressed by the indefinite quadratic augmented matrix inequality (12), in which the stationary point of corresponds to a projection in the following Krein subspace:withDenote , . The stationary point of the indefinite quadratic form (12) corresponds to the projection of into the Krein subspace spanned by . Let be the projection of into , and let be . Then is an orthogonal basis of , and the projection of into the Krein subspace is given byThe projection of into iswhereDenote ,; thenThe projection of in (17) corresponds to a stationary point of the indefinite quadratic form in (12), and the value of at this stationary point isin which Due to the fact that can be expressed as the form in (23), can also be given by (24):Here, According to [19, 20], is the minimum of if and only if and have the same inertia. Considering the block triangular factorization of as shown in (23), the sufficient condition of the minimum is . Therefore, a choice of to ensure is ; then the estimate of the signal to be estimated isin whichThe existing condition of this filter is that and have the same inertia index.
The Riccati equation is given by
Remark 2. Given a positive scalar , the value of and can be obtained in each fusion period. Then compare the inertia index (the number of positive eigenvalues) of and . If they have the same inertia index, the fusion filters in Theorem 1 are applied to the corresponding fusion period. Otherwise, the value of should be changed, and the fusion filter needs to be resolved for the changed performance index function.
In Theorem 1, a robust finite horizon fusion filtering algorithm is proposed based on the centralized fusion strategy. The estimate of the signal to be estimated cannot be obtained until all measurements sampled by different sensors arrive at the fusion center, by a high-dimensional operation. Obviously, the real time property of the centralized fusion methods is usually lost to some extent in this fusion filtering process.
Motivated by this situation, in the next subsection, an equivalent real time robust finite horizon fusion filtering algorithm is also proposed for the time-varying -sensor system with uncertain parameter, on the basis of the sequential fusion strategy.
3.2. Sequential Robust Finite Horizon Fusion Filtering Algorithm
Without loss of generality, assume that the arrival sequence of the measurements is just the sequence of the sensors; namely, the measurements arrive at the fusion center in the sequence . These measurements could be dealt with sequentially in the fusion center, according to the sequential fusion strategy. Denote the corresponding system state of by ; then one gets
Lemma 3. The performance index function shown in (9) can also be expressed as the following set of quadratic inequalities:
Theorem 4. For the augmented matrix system shown in (11), the following sequential robust finite horizon fusion filter can be given to satisfy the performance index functions (30a), (30b), and (30c):in whichThe existence condition of this robust fusion filter is that and have the same inertia index.
An analytical proof of this theorem is given in the remainder of this section.(I)When arrives at the fusion center, the corresponding subsystem is
The corresponding performance index function is (30a), the stationary point of which corresponds to a projection in the following Krein subspace:with
The stationary point of (30a) corresponds to the projection of in the Krein subspace spanned by . Denote the projection of in by , and the error . Obviously, is an orthogonal basis of . Therefore, the projection of in is given by
The projection of in is given bywhere
One gets
corresponds to the stationary point of (30a), the value of at which point iswhere ,.
Because ,.
The estimate of the signal to be estimated is given byin which
Because , the corresponding Riccati equation is given by(II)When arrives at the fusion center, the corresponding subsystem is
The corresponding performance index function is (30b), the stationary point of which corresponds to a projection in the following Krein subspace:with
The stationary point of (30b) corresponds to the projection of in the Krein subspace spanned by . Denote the projection of in as , and . Obviously, is an orthogonal basis of . Therefore, the projection of in is given by
The corresponding projection of is given byin which Then,
corresponds to the stationary point of (30b), the value of at which point iswhere ,. Therefore, .
Then the estimate of the signal to be estimated is given by
Because , the corresponding Riccati equation is given by(III)When arrives at the fusion center, the corresponding subsystem is
The corresponding performance index function is (30c), the stationary point of which corresponds to a projection in the following Krein subspace:with
The stationary point of (30c) corresponds to the projection of in the Krein subspace . Denote
Then is an orthogonal basis of . And we can obtain the projection of in as
The corresponding projection of is given by
Denote ; then one gets
corresponds to the stationary point of (30c), the value of at which point isin which